The positive integers P, Q, P - Q, and P + Q are all prime

Question

The positive integers P, Q, P - Q, and P + Q are all prime numbers. Which of the following must be prime?

I. Q + 1
II. P + 2Q
III. 3P + 2Q

(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

rajdeep212003 · Answer

Given:
P, Q, P−Q, and P+Q are all prime

Since P−Q is prime,
difference between two primes must be prime → this only works when Q = 2
(otherwise difference becomes even > 2, not prime)

So take:
Q = 2

Now find P such that:
P−2 and P+2 are both prime

This works for:
P = 5 → (3, 5, 7 all prime)

---

Check options:

I. Q + 1 = 2 + 1 = 3 → prime ✔️

II. P + 2Q = 5 + 4 = 9 → not prime ❌

III. 3P + 2Q = 15 + 4 = 19 → prime ✔️

---

Final Answer:
I and III only

---

Hope this helps.
Feel free to tag me if you want to go through it.

– Rajdeep

HuanTrinh · Answer

Q is surely 2

However, I am wondering if there is any logical explaination rather than trying to find P. Obviously if you remember the list of prime numbers, there is no such prime number x that x - 2 and x + 2 are both prime except 5.

Correct answer is surely D. I am just trying to find a more optimized way to solve this. Really good question!

DeepGOO225 · Answer

Can we not take the value of p and q as like difference between the two is 2 like 7 and 5 , 5 and 3 ,etc so in this case solution will change

HarshavardhanR · Answer

P, Q, P - Q, P + Q --- are all prime numbers.

Inference 1:   Q = 2.

Because: 
- If P and Q were odd prime numbers, then P+Q would be odd + odd = even number greater than 2. In other words, there is no way P + Q will be prime in such a case.
- Therefore, among P and Q, one of them is THE even prime number i.e., 2.
- P - Q is also prime. If P = 2, then P - Q would be a negative number. Prime numbers are not negative. Therefore, P is not = 2. 
- Hence, Q = 2.

P - Q, P, and P + Q are prime.

So, knowing that Q = 2,

P - 2, P, and P + 2 are prime. Also, remember that P is an odd prime number.

(P - 2, P, P + 2)

(1,3,5) - not valid. Because 1 is not prime.
  (3,5,7) - valid.   
(5,7,9) - not valid. Because 9 is a multiple of 3.
(9,11,13) - not valid. because 9 is a multiple of 3. 
(11,13,15) - not valid. because 15 is a multiple of 3. 
And so on.

Inference:  Barring (3,5,7), no other set of (P-2, P, P+2) works.

It is actually easy to see why.

Think:  (P - 2, P, P + 2) is essentially 3 consecutive odd numbers.

In any 3 consecutive odd numbers, one number will be a multiple of 3.

This is also true for (3,5,7) but here, given that 3 is a prime number, this was not an issue.

For any other set,  any 3 consecutive odd numbers (P-2, P, P+2) cannot all be prime numbers,  because in any 3 consecutive odd numbers,  one number will be a multiple of 3 . So, here, even if P is prime, either P+2 or P-2 will be a multiple of 3, and therefore, not prime.

Bottom Line : We have a unique answer set.

Q = 2. P = 5. P - 2 = 3. P + 2 = 7.

(I) Q + 1 = 2 +1 = 3. Must be prime.
(II) P + 2Q = 5 + 2(2) = 9. Not prime.
(III) 3P + 2Q = 15 + 4 = 19. Definitely Prime.

Answer:   I and III only. Choice D .

---
Harsha

egmat · Answer

Hi HuanTrinh,

There is, and it comes down to a single fact about  3 . No prime list needed.

You already have Q =  2 , so the real question is the one you named: why is  5  the only prime P for which P -  2  and P +  2  are also prime?

Here is the fact that settles it:  among any three numbers spaced 2 apart, one is always a multiple of 3.

Why? Every whole number is one of exactly three kinds - a multiple of  3 , one more than a multiple of  3 , or two more than a multiple of  3 . There is no fourth kind. So take P and check each case:

- P is a multiple of  3 . Then P itself is the one.
- P is one more than a multiple of  3  ( 7 ,  10 ,  13  ...). Then P +  2  is the multiple of  3 . Check:  7  gives  9 ,  10  gives  12 .
- P is two more than a multiple of  3  ( 5 ,  8 ,  11  ...). Then P -  2  is the multiple of  3 . Check:  5  gives  3 ,  8  gives  6 .

Every case lands on a multiple of  3 .  Therefore  one of P -  2 , P, P +  2  is divisible by  3 , always.

But all three have to be prime, and the only prime divisible by  3  is  3  itself. So one of them must literally equal  3 . Only three placements to try:

- P +  2  =  3 , so P =  1 . Not prime.
- P =  3 , so P -  2  =  1 . Not prime.
- P -  2  =  3 , so P =  5 . Gives  3 ,  5 ,  7  - all prime.

P =  5  is forced, not remembered. From there: I gives  3  (prime), II gives  9  (not prime), III gives  19  (prime).

Answer: D

The positive integers P, Q, P - Q, and P + Q are all prime

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The positive integers P, Q, P - Q, and P + Q are all prime

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